1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Limits on an Integral of a semi-circle

  1. May 28, 2012 #1
    1. The problem statement, all variables and given/known data

    A question asks to calculate the integral over the region R given by:

    x^2 + y^2 <= 4
    0 <= y <= 2

    Which would be the upper half of a circle of radius 2 centred on the origin.

    The integral is done in the book I have and the limits of x are given as -2 to 2, which I can understand.

    Though the limits for y are given as: 0 to (4 - x^2)^0.5

    I can see that they have obtained this limit from rearranging the first part of the region R.

    BUT, why is the limit for y not 0 to 2. Or alternatively, if what they have done is correct, why is it not equally valid to state the limits for x are: 0 to (4 - y^2)^0.5

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. May 28, 2012 #2
    When integrating over a region, we need to hit each point in the region exactly once, and no other points.

    So your recipe is, for each x betwen -2 and 2, (hold x fixed and) integrate over y=0 to 2. But then you would hit all points in a rectangle, not the semicircle.

    You'll hit each point in the semicircle once if, for each x between -2 and 2, you go from y= 0 to the upper circle.

    If this seemed to make sense, here's some exercises to check that you understand.

    Try switching order of integration, that is, for each y in between (which values?), let x runf from where to where, thus hitting each point in the region.

    Another exercise. In the original limits, hold x fixed and let y vary, why do we take x goes from -2 to 2. Why not a smaller or larger region, what exactly would be wrong with that, how geometrically or by what type of value would it change the answer. (Hint: integration works if we hit each point in region exactly once, and no others.)
  4. May 28, 2012 #3


    User Avatar
    Science Advisor
    Homework Helper

    Hi ZedCar! :smile:
    It is, if the y limits are -2 to 2. :smile:

    You can have vertical slices of thickness dx and height √(4 - x2)

    or horizontal slices of thickness dy and width √(4 - y2). :wink:
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook